Inclusion-Exclusion Principle/Examples/3 Events in Event Space

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Examples of Use of Inclusion-Exclusion Principle

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $A, B, C \in \Sigma$.


\(\ds \map \Pr {A \cup B \cup C}\) \(=\) \(\ds \map \Pr A + \map \Pr B + \map \Pr C\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \map \Pr {A \cap B} - \map \Pr {B \cap C} - \map \Pr {A \cap C}\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \map \Pr {A \cap B \cap C}\)


The Inclusion-Exclusion Principle is applicable for an additive function on an algebra of sets.

We have that $\Sigma$ is a $\sigma$-algebra.

Hence by definition, $\Sigma$ is an algebra of sets which is closed under countable unions.

Hence, a fortiori, $\Sigma$ is an algebra of sets.

We have by definition of probability measure, that $\Pr$ is an additive function fulfilling certain conditions.

The result then follows as a special case of the Inclusion-Exclusion Principle.